Lecture 11
Set Builder Notation

Definitions:

Set Builder Notation: Notation that can easily be used to express sets containing an infinite number of elements.

How to “read” set builder notation.

Say, “The set of all x…” when seeing: { x
Then say, “such that…”, when seeing: |

For instance:

{ x | x > 0 and x Integer Numbers } is read, “The set of all x, such that x is greater than zero and x is an element of the set of Integer Numbers.”.

Note:
{ x | x > 0 and x Integer Numbers } = { 1 , 2 , 3 , … }

For instance:

{ x | x < - 3 and x Integer Numbers } is read, “The set of all x, such that x is less than negative three and x is an element of the set of Integer Numbers.”.

Note:
{ x | x < - 3 and x Integer Numbers } = { … , - 5 , - 4 , - 3 }

For instance:

{ ab | a = - 1 and b Natural Numbers } is read, “The set of all a times b such that, a equals negative one and b is an element of the Natural Numbers.”.

Note:
{ ab | a = - 1 and b Natural Numbers } = { … , - 3 , - 2 , - 1 }

Objectives:

· To use Set Builder notation to express sets.

· To convert Set Builder notation to standard set notation by listing elements.

· To use Set Builder notation to define Rational Numbers.

Examples:

1. Rewrite { x | x > 0 and x Integer Numbers } in standard set notation by listing elements.

{ x | x > 0 and x Integer Numbers } = { 1 , 2 , 3 , … }

2. Rewrite { x | x Integer Numbers } in standard set notation by listing elements.

{ x | x Integer Numbers } = { … , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , … }

3. Rewrite { x | x is even and x is prime } in standard set notation by listing elements.

{ x | x is even and x is prime } = { 2 }

Exercises:

1. Rewrite { x | x > 1 and x Integer Numbers } in standard set notation by listing elements.

2. Rewrite { x | x Natural Numbers } in standard set notation by listing elements.

3. Rewrite { - x | x Natural Numbers } in standard set notation by listing elements.

4. Rewrite { x | x Whole Numbers } in standard set notation by listing elements.

5. Rewrite { x | x > -1 and x < 1 and x Integer Numbers } in standard set notation by listing elements.

6. Rewrite { x | x > -1 and x < 1 and x Whole Numbers } in standard set notation by listing elements.

7. Rewrite { 2n | n Integers Numbers } in standard set notation by listing elements.

8. Rewrite { 2n + 1 | n Integers Numbers } in standard set notation by listing elements.

Selected Solutions:

1. Rewrite { x | x > 1 and x Integer Numbers } in standard set notation by listing elements.

{ x | x > 1 and x Integer Numbers } = { 2 , 3 , 4 , … }

2. Rewrite { x | x Natural Numbers } in standard set notation by listing elements.

{ x | x Natural Numbers } = { 1 , 2 , 3 , … }

3. Rewrite { - x | x Natural Numbers } in standard set notation by listing elements.

{ - x | x Natural Numbers } = { … , - 3 , - 2 , - 1 }

4. Rewrite { x | x Whole Numbers } in standard set notation by listing elements.

{ - x | x Natural Numbers } = { 0 , 1 , 2 , 3, … }

5. Rewrite { x | x > -1 and x < 1 and x Integer Numbers } in standard set notation by listing elements.

{ x | x > -1 and x < 1 and x Integer Numbers } = { 0 }

6. Rewrite { x | x > -1 and x < 1 and x Whole Numbers } in standard set notation by listing elements.

{ x | x > -1 and x < 1 and x Whole Numbers } = { 0 }